New Trends in Statistical Physics

The following sections are included:

• Preface

• Contents

In the experimental devices used for Bose–Einstein condensation the gas is trapped resorting to magneto–optical methods. In other words, the concept of volume of the corresponding container (as usually found in any text book on statistical mechanics) becomes in these experiments a rather vague parameter. In the present work we deduce the concepts of volume and pressure (as a function of the variables associated to the trap) that allow us to define the corresponding ensembles. Additionally, the issue of the fluctuations in the number of particles is addressed.

Shockwaves provide a useful and rewarding route to the nonequilibrium properties of simple fluids far from equilibrium. For simplicity, we study a strong shockwave in a dense two-dimensional fluid. Here, our study of nonlinear transport properties makes plain the connection between the observed local hydrodynamic variables (like the various gradients and fluxes) and the chosen recipes for defining (or “measuring”) those variables. The range over which nonlocal hydrodynamic averages are computed turns out to be much more significant than are the other details of the averaging algorithms. The results show clearly the incompatibility of microscopic time-reversible cause-and-effect dynamics with macroscopic instantaneously-irreversible models like the Navier-Stokes equations.

We consider a system of particles subjected to a uniform external force E and undergoing random collisions with “virtual” fixed obstacles, as in the Drude model of conductivity.¹ The system is maintained in a nonequilibrium stationary state by a Gaussian thermostat. In a suitable limit the system is described by a self consistent Boltzmann equation for the one particle distribution function f. We find that after a long time f(v, t) approaches a stationary velocity distribution f(v) which vanishes for large speeds, i.e. f(v) = 0 for ∣v∣ > v_max(E), with v_max(E) ≃ ∣E∣⁻¹ as ∣E∣ → 0. In that limit f(v) ≃ exp(−c∣v∣³) for fixed v, where c depends on mean free path of the particle. f(v) is computed explicitly in one dimension.

The soft nature of liquid crystals induces highly nonlinear couplings between the optical field and the director. This nonlinear dynamics makes it possible to have solitons of the optical fields. In this work the conditions under which a new type of solitons (embedded solitons, ES) may exist in liquid crystals waveguides is discussed. To describe this new type of solitary waves, we first review some basic concepts of the hydrodynamics of liquid crystals and then discuss how the ES were discovered and what is the mechanism that accounts for their existence. We use a model for a cylindric nematic waveguide and write the coupled basic equations for the electromagnetic modes in the guide and for the orientation field. Then we use a simplified multiple scale technique to derive the structure of the basic equations which describe the evolution of light pulses in liquid crystals at different scales. We study the ES of the complex modified Korteweg de Vries equation (cmKdV) and use a variational method to explain the stability of its solitons.

The solution of the Fokker-Planck-Kramers equation for a charged Brownian particle in an electromagnetic field, is taken as a starting point to establish some fluctuation relations for the configuration and velocity space probability densities. These fluctuation relations are related with the work done by the external fields along the forward and backward trajectories. The electric field is in general time dependent and the magnetic field a constant. As a result we have found two kinds of fluctuation relations, both in configuration and velocity spaces. The first one is called the barotropic fluctuation relation and the second, and more interesting, measures how the magnetic and the electric fields couple to give place to the Hall fluctuation relations.

A study of the binary mixture liquid-vapor azeotrope within the frame of basic equilibrium thermodynamics is presented. In the introduction we review the defining properties of the azeotrope. In a second section we demonstrate a new general property that links, in the limit of the azeotrope, the pressure as a function of the liquid composition and the pressure as a function of the vapor composition, both computed at a given value of temperature, to the composition of the vapor as a function of the liquid composition at that same temperature and, analogously for the temperature as a function of the liquid composition and of the vapor composition for a fixed value of the pressure. We also give a demonstration of the fact that whenever the azeotrope occurs as a maximum in the pressure vs composition coordinates it will be a minimum in the temperature vs composition coordinates and viceversa. In a third section we construct a polynomial model for the above mentioned coexistence pressure functions of composition of liquid, and of vapor, based on the requisites that are to be met by the pressure at the extremes of the pure components and at the azeotrope.

The problem of a zero-mean oscillatory flow of a Newtonian fluid between infinite parallel plates with thermal boundary conditions of the third kind is considered. With the analytic solutions for the velocity and temperature fields at hand, the local and global time-averaged entropy production are computed. The consequences of having different convective heat transfer coefficients in each plate are assessed for this problem and some conditions that lead to entropy generation minimization are determined.

The assumption of a zero-point energy to derive Planck's law within a continuous (non-quantum) perspective and a thermodynamic context has been studied previously. However, some of the arguments involved in such derivations, even though physically sustained, lack of a demonstrative character. In this paper we address these deficiencies. In doing so, we complete a sound demonstration showing that Planck's law —and hence the ensuing discrete description— is a necessary consequence of the existence of the zero-point zero-point energy. We also disclose the relevance of Wien's law for reaching this conclusion.

In this work a theoretical method is presented instead of the usual numerical method to analyze the Newton's endoreversible engines. The power output function and the ecological function are expressed in dimensionless form, a functional relationship between these functions is found. It essentially shows that the ecological function is proportional to the re-scaled power output function. Using this result two additional functional relationships are found, the first between the upper operational temperatures: and , and the second between the efficiencies η_P and η_E.

Using two physical facts on the heat flux and on the efficiency, the general analytical behavior of the dimensionless power output function is obtained, and combining it with the functional relationship the analytical behavior of the ecological function is obtained. Using these analytical results and the functional relationships between the upper operational temperatures, and between the efficiencies, it is shown that for Newton's endoreversible engines the following inequalities: and η_E > η_P are always satisfied.

It is shown that from the operational point of view, the functional relationship reduces the amount of numerical calculation. A unified procedure to obtain the upper operational temperature and the efficiency for a Newton's endoreversible engine, upon using either the maximum power output or the maximum ecological function criteria is presented.

The concentration change due to diffusion between two chambers connected by a conical capillary tube is approached by a microscopical model, by means of the use of propagator functions, G_ji, which state the probability of found a particle at time t in chamber j, given that the particle was in chamber i at time t = 0. The description of the propagator functions involved the equilibrium probabilities and relaxation functions which describe the way the system reach the equilibrium state. The latter functions had to be written down in terms of the fluxes of particles escaping from the capillary, f_tr,i and f_r,i, named the translating and returning fluxes from chamber i. To obtain the fluxes, the Fick-Jacobs' equation was solved inside the tube, subject to radiative boundary conditions, in the Laplace's space, then used these solutions to write down the Laplace transforms of the propagator functions. Monte Carlo computational simulations were performed to complement the analytical solution previously obtained by finding the appropriate model to describe the effective diffusion in the capillary tube, in the interval of geometrical parameters accessible to actual biological systems. Our results show that this model could has serious applications in modelling the diffusion between the cell's surroundings and its interior trough a varying cross-section channel, as those specific to the potassium ions, and contribute to understand key features of those systems. Additionally, the simulated data yielded two noticeably results: the non-equivalent behavior of diffusion trough the conical tube went trough in opposite directions, and the inability of present theoretical models to completely describe the observed patterns.

Gene transcription or Gene Expression (GE) is the process which transforms the information encoded in DNA into a functional RNA message. It is known that GE can occur in bursts or pulses. Transcription is irregular, with strong periods of activity, interspersed by long periods of inactivity. If we consider the average behavior over millions of cells, this process appears to be continuous. But at the individual cell level, there is considerable variability, and for most genes, very little activity at any one time. Some have claimed that GE bursting can account for the high variability in gene expression occurring between cells in isogenic populations. This variability has a big impact on cell behavior and thus on phenotypic conditions and disease. In view of these facts, the development of a thermodynamic framework to study gene expression and transcriptional regulation to integrate the vast amount of molecular biophysical GE data is appealing. Application of such thermodynamic formalism is useful to observe various dissipative phenomena in GE regulatory dynamics. In this chapter we will examine at some detail the complex phenomena of transcriptional bursts (specially of a certain class of anomalous bursts) in the context of a non-equilibrium thermodynamics formalism and will make some initial comments on the relevance of some irreversible processes that may be connected to anomalous transcriptional bursts.

As a consequence of the fundamental importance of peptides and proteins in physiological events, the interest in peptide synthesis as well as in their characterization has increased exponentially in recent years. Although the vast majority of natural peptides and proteins are constituted by α-amino acids, recent studies have shown that the incorporation of β-amino acids, instead of α-amino acids, in peptides significantly modifies their biological activity and increases their hydrolytic stability. As a consequence, inclusion of β-amino acid residues in peptides increases significantly the potential application of the resulting unnatural peptides. The present compilation is intended to provide an overview of the more significant advances achieved in the synthesis and applications of β-peptides in chemistry, biology, and medicine that it is being carried out in Mexico applying modern synthetic methods.

Different laser light sources are used to selectively excite the various natural fluorophores present in mononuclear live cells. The radiation characteristics using a laser-induced fluorescence technique together with an appropriate tissue preparation permitted an improved spectral resolution. The NADH spectrum in phosphate buffered saline solution was very closely fitted with a double Gaussian that was used as the reference spectrum for the intra cellular experiments. The NADH molecules within the cells provide the dominant contribution to the relatively narrow band fluorescence spectrum obtained with this procedure under 355 nm Nd:YAG laser excitation. The deconvolution of the fluorescence curve allowed for the direct calculation of the NADH bound/free ratio. This dimensionless quantity, estimated at 1.35 with this method, is an important indicator of the energetic metabolic state of the cells.

A new Bloch electron dynamics in which the “electron” (“hole”) wave packet, called simply “electron” (“hole”), has a size besides a charge, is introduced to describe the electrical conduction in graphene based on a rectangular unit cell for the graphene's honeycomb lattice. The conduction of a single-wall nanontube (SWNT) is semiconducting or metallic depending on whether the pitch contains an integral number of carbon hexagons along the tube axis or not. The low-temperature residual conductivity of the semiconducting SWNT is shown to arise from the Cooper pairs formed by the phonon exchange attraction. A quantum statistical theory is presented supporting a superconducting state with an ultrahigh critical temperature (1275 K) in the multi-walled nanotubes reported by Zhao and Beeli [Phys. Rev. B 77, 245433 (2008)].

Fractional quantum Hall systems are the prime example of strongly interacting electron systems. At very high values of the applied magnetic field, there is only one relevant energy scale, the Coulomb interaction V_c ≈ e²/λ, where λ is the magnetic length. Conventional many-body perturbation theory is inapplicable to such systems. Insight into the nature of the correlations was first given by Laughlin. This important contribution together with contributions by Haldane, Halperin, and Jain are reviewed in light of some rigorous mathematical theorems in an attempt to gain further insight into the correlations.

We revisit the Gibbs-di Marzio and di Marzio-Dowell models for the calorimetric properties of polymers in the glass transition region. We find an expression similar to the one given by di Marzio-Dowell for the specific heat that fits very well with the experimental data reported in the ATHAS database, and using these results we find the logaritmic shift factor that fits well with the newest experimental data available for the relaxation times.

Incorporating relativity principles with those of kinetic theory is important not only to understand their theoretical foundations but to actually interpret current high energy and astrophysical phenomena. In this work we study the relativistic Lorentz gas: a binary mixture composed of light relativistic particles that diffuse in a background gas made up of heavy non-relativistic particles, so that m ≪ m_G and n ≪ n_G, with suffix _G indicating the non-relativistic component whereas quantities without index indicate the relativistic one; m and n stand for the rest masses and particle densities, respectively. We start by considering the relativistic Boltzmann equation for the relativistic component while component _G is assumed to be in equilibrium. As it is well known the transport coefficients can be investigated by making approximations for the collision term in Boltzmann's equation, in particular by replacing it with a Fokker-Planck differential operator in which diffusion and friction coefficients appear directly. In the case of the relativistic Lorentz gas we consider we get an extra term proportional to the distribution function besides the usual Fokker-Planck contribution. Our analysis can be considered complementary to a previous one in which, where the background gas was relativistic and the diffusing component was the non-relativistic one

It is shown by means of a simple analysis that the linearized system of transport equations for a relativistic, simple fluid at rest obeys the antecedence principle, which is often referred to as causality principle. This task is accomplished by examining the roots of the dispersion relation for such a system. This result is important for recent experiments performed in relativistic heavy ion colliers, since it suggests that the Israel-Stewart like formalisms may be unnecessary in order to describe relativistic fluids.

The complete structure of geodesics in fast Kerr and Kerr–(anti-)de Sitter space–times are analyzed and classified. The analysis is based on the determination of the zeros of the polynomials underlying the equation of motion for the altitudinal and the radial coordinates.

From a general cyclic symmetric stationary metric of the 2 + 1–gravity a shortcut in the derivation of the three families of cyclic solutions is given. There arise two branches for the BTZ black hole solution, and additionally the Coussaert-Henneaux solution and the cyclic SO(2) × SO(2) cosmological metric.

Quasi–local (QL) scalar variables are introduced for spherically symmetric LTB models in order to examine the “back–reaction” term Q in the context of Buchert's scalar averaging formalism. Since QL scalars defined as functionals become weighed averages that generalize the standard proper volume averages, we are able to provide rigorous proof that back–reaction is positive for (i) all LTB models with negative and asymptotically negative spatial curvature, and (ii) models with positive curvature decaying to zero asymptotically in the radial direction. Qualitative, but robust, arguments are given in order to prove that generic LTB models exist, either with clump or void profiles, for which an “effective” acceleration associated with Buchert's formalism can mimic the effects of dark energy.

The following sections are included:

• Author Index

• Subject Index